As others had before him, Crommelin and his staff returned to their calculations after the comet passed and assessed the accuracy of their results. Taken as a group, the calculations of 1835 fell within sixteen days of the true date of perihelion. The single 1910 computation was within two days, sixteen hours, and forty-eight minutes of the correct value, an improvement by a factor of about five. Crommelin claimed that the method of mechanical quadratures, “if pressed to the extreme accuracy of which it is capable, will give results of higher degrees of accuracy than any previously published method of dealing with this comet.” He attempted to demonstrate the validity of this claim by making a more refined calculation of the comet’s orbit. After expending much effort on the computation, he found that his revised work was no better than his original estimate. He did not withdraw his claims for mechanical quadratures but instead speculated that something other than mathematical accuracy was causing the discrepancy. He suggested that some additional force, an interstellar drag, was slowing the progress of Halley’s comet.12
After the comet had again been lost to view in the outer reaches of the solar system, just a few years after Crommelin had complained that no was one engaged in computing the orbit of Halley’s comet, the English mathematician Edmund Whittaker (1873–1957) wrote that “there has been a great awakening of interest in [computation]; and it is now included in the syllabus for the [British Civil Service] Examination.” The calculation of a comet’s orbit might no longer pose an opportunity for new discoveries, but there were other fields that looked to scientific calculation as a way of providing detailed, precise answers. Whittaker observed that a knowledge of calculation was “required by workers in many different fields—astronomers, meteorologists, physicists, engineers, naval architects, actuaries, biometricians, and statisticians.”13 In 1913, he was a professor at the University of Edinburgh, in Scotland, and was in the process of forming a laboratory devoted to the topic of computation. Most of his efforts were devoted to cataloging the methods of calculation and to creating new techniques that might be applied to broad classes of problems, just as the method of mechanical quadratures might be applied to problems other than those of comet orbits.
Whittaker had been trained as an astronomer and, for a time, had served as the Astronomer Royal of Ireland. Like other astronomers before him, Whittaker had been drawn to the mathematics of insurance and, according to his biographer, had “been influenced by his friendship with the great actuaries of the period.”14 His laboratory borrowed from actuarial practice, a practice that demanded that computers follow calculating plans to the letter and that they record their intermediate values in ways that would allow others to verify their work. Whittaker gave detailed recommendations for every aspect of computing. He told computers to write numbers in pairs, for it is “found conducive toward accuracy and speed,” and argued that “every computation should be performed with ink in preference to pencil; this not only ensures a much more lasting record of the work but also prevents eyestrain and fatigue.” He felt that all computation should be done on specially prepared paper that was “divided by a faint ruling into 1/4 squares, each of which is capable of holding two digits.” His recommendations even included the desks that should furnish the computing room. He stated that the desks “used in the mathematical laboratory of the University of Edinburgh are 3′ 0″ wide, 1′ 9″ from front to back and 2′ 6½″ high. They contain a locker, in which computing paper can be kept without being folded, and a cupboard for books, and are fitted with a strong adjustable book-rest.”15
Beginning with his first writings on the subject, Whittaker was a strong advocate for direct numerical calculation. Many scientists still relied on graphical means for calculation. They would find a way of constructing some two-dimensional shape that had an area equal to the desired result. Once the shape was finished, the correct answer could be determined by measuring the area of the shape. Edmund Halley had used such a method when he first looked at the orbits of comets. Just as Halley had abandoned this technique, Whittaker advised others to do the same. He wrote that at his Edinburgh laboratory “graphical methods have almost all been abandoned, as their inferiority has become evident, and at the present time the work of the Laboratory is almost exclusively arithmetical.”16
The staff of the Edinburgh Mathematics Laboratory worked on a variety of problems, but they devoted most of their efforts to the construction of mathematical tables, to computing values of complicated mathematical functions that appeared in many different forms of scientific or engineering endeavor. Simple examples of mathematical tables include the familiar pages of logarithms, sines, and tangents and the probability functions of Karl Pearson and his Biometrics Laboratory. The Edinburgh computers tried their hands at such tables, but they were most interested in tabulating the Bessel function. The Bessel function was named for Friedrich Wilhelm Bessel (1784–1846), a German scientist who had been known as a mathematician and as the director of an astronomical observatory in Königsberg. “Many mathematicians, usually working in celestial mechanics, arrived independently at the Bessel function,” wrote the historian Morris Kline.17 Bessel wrote the first systematic treatment of this function in 1824 as an outgrowth of his study of planetary motion. The task of tabulating this function was substantial, as it appeared in two different forms and behaved in a wide variety of ways. In the late nineteenth century, it emerged as one of the great problems for scientific computers. The orbit of Halley’s comet was computed once every 75 years and then packed away for another generation. The Bessel function found applications in ever-broadening fields of science. It was useful to anyone studying problems of vibration, including vibrating drums, vibrating air, and vibrating electrical signals. It also could be applied in the study of heat and the diffraction of light.18 As scientific computing moved from Greenwich to Edinburgh, from Halley’s comet to the Bessel function, it slipped, at least slightly, from the grasp of astronomers and was picked up by scientists who were studying phenomena that were much closer to the earth.
CHAPTER NINE
Captains of Academe
War rolled swiftly up the beach and washed the sands where Princeton played. Every night the gymnasium echoed as platoon after platoon swept over the floor and shuffled out the basketball markings.
F. Scott Fitzgerald, This Side of Paradise (1920)
DURING THE LAST DAYS of July 1914, in the final hours of peace, the European powers positioned themselves for the impending conflict. Germany prepared to march its army through the supposedly neutral country of Belgium. The French hurried to throw their military might between the advancing troops and Paris. The English, perceiving that they had interests on the Continent, organized an expeditionary force to send into the fray. Karl Pearson, the great admirer of German culture, found himself caught on the European side of the English Channel. He hurried home to London on the first day of the conflict and declared that the needs of his country were more important than his personal ambition or his love of science. “On August 3, 1914,” he wrote, “I at once put the whole Laboratory staff at the service of any [British] Government department that was in need of computing or statistical aid.”1
In 1914, the Biometrics Laboratory employed a staff of ten computers, four men and six women. The military could have utilized all ten. They would have enlisted the men qualified for service and made them navigators and surveyors. The women, and those men who were unfit for military duty, would have been given jobs as clerks or engineering assistants. Pearson argued that the group should remain intact and under his control. “The Laboratory can do far better work nationally as a whole than scattered, as it is trained to work together.”2
At first, he was willing to accept relatively menial assignments for his computers. Beginning that fall, the Biometrics Laboratory “provided weekly some 500 or more graphs showing the state of unemployment of insured and of uninsured trade men and women.” The work required no advanced mathematics and was relatively straightforward, even though it
kept the computers on a strict production schedule. “Six and sometimes eight series of these graphs were kept running and carried to date each week,” Pearson reported. The staff worked “without a break through all the vacations up to July [1915],” balancing the requirements of the statistical reports with the demands of Pearson’s biometrical research.3
When the laboratory staff returned from the summer vacation of 1915, Pearson learned that half of his workers were not satisfied with the role that they were playing in the war effort. The conflict still retained its heroic potential, its romantic promise of bold actions and daring deeds. On those days when the wind blew from the southeast, the more alert residents of London could hear the report of cannon and the muffled thud of exploding shells. Three of the male computers had tried to enlist in the army with the hope of serving on the front. All three were rejected by the recruiting office, but each had found a position in industries that were supplying the military. An equal number of female computers had also left the lab, one to serve in a hospital, one to teach, and one to rejoin her family.4 Pearson, conscious that he would have to train and recruit a new staff, decided to temporarily withdraw from war work.
On September 9, the war reached the Biometrics Laboratory. “We are all congratulating ourselves that we have seen a Zeppelin at last,” wrote one of the female computers. The Zeppelin had crossed the sea from Germany and followed railroad tracks into London. “I was coming home in a tram just before 11 PM,” she added, “when the driver called out that there was a Zep.” She got out of the tram and began walking home, keeping an eye on the big, hulking shape in the night sky. “Nobody obeyed the Instructions to seek shelter. We could see the flashes from the anti-aircraft guns but they all went very wide of the mark.” The airship was looking for the Charing Cross railroad station, which was situated on the river Thames, but it dropped a bomb near the university “and nearly every window is smashed and numerous shops were destroyed.” The computer reported that the population seemed to regard the event as an adventure rather than as a threat, claiming that the “whole of London is in a state of subdued excitement.”5
Pearson claimed that he was able to face the bombings with resolution. “I just went about my usual tasks,” he wrote after the war: “I made belief that it was nothing.”6 By December, he felt that his computers were ready to resume war work. This time they created shipping reports, “preparing graphs dealing with the tonnage required for various imports for the use of committees controlling these matters.”7 He reported that the work occupied all of the computers’ time, but at least one of his computers was still doing biometric computations. Pearson continued with his statistical research but spent increasing amounts of his time doing analyses for aircraft factories. The aviation industry was only a decade old, and its engineers had much to learn about structure, motion, lift, and drag. Most of the problems undertaken by Pearson involved the flexing of structural parts: propellers, wing struts, airframes. The mathematics of this analysis was somewhat tricky, but Pearson had studied the subject before he had become interested in mathematical statistics. The work brought him into direct contact with the military and introduced him to the big mathematical problem of the war: the calculation of bomb and shell trajectories, a subject known as mathematical ballistics.8
Mathematical ballistics lay behind the most brutal weapons of the war. It allowed artillery crews to aim their guns at distant targets, mortar crews to lob gas-filled shells from behind the protection of a hill, moving Zeppelins to bomb stationary structures, and defense gunners to destroy invading aircraft. It had a history as old and venerated as the history of mathematical astronomy, and it had a problem every bit as difficult as the three-body problem of Halley’s comet: the problem of modeling the atmospheric drag on the flying shell. This problem had been first encountered in the fifteenth-century cannonball experiments of Galileo Galilei (1564–1642). Galileo argued that the air had no effect upon the motion of the balls and concluded that trajectories were graceful parabolas through the sky.9 Newton placed Galileo’s analysis on a more formal foundation, but like his predecessor, he assumed that the cannonball was moving in vacuo. By the early eighteenth century, scientists had discovered that air resistance had a substantial impact upon ballistics trajectories, and they had also found it to be a complicated phenomenon. As a projectile neared the speed of sound, it created shock waves that dissipated energy and greatly increased the drag. For extremely high velocities, such as those achieved by projectiles as they left the barrel of a gun, the nature of the drag changed again. These variations thwarted any attempt to make a simple calculation of a trajectory. Scientists of the mid-nineteenth century could compute a trajectory only by using methods similar to those that had been used by Alexis Clairaut for Halley’s comet. They would track a projectile along an idealized path and adjust its position to account for the drag.10
A clearer understanding of air resistance began to emerge in the 1860s, as military engineers began to amass a large collection of data. The first of this data came from pendulum tests. Gunnery crews would fire a shell at a large pendulum and measure the displacement of the bob. From these measurements, the engineers could determine the velocity of the shells and, ultimately, the atmospheric drag. By 1870, the armies of Prussia, Great Britain, France, Russia, and Italy had all estimated the nature of drag. These estimates took the form of graphs rather than mathematical expressions. No simple expression described the relationship between the velocity of a shell and the drag. The French estimate, usually called the Gâvre function, after the French proving ground, was generally considered to be the most reliable of the time. It was used by an Italian professor, Francesco Siacci (1839–1907), to create a simple means of computing trajectories.
Siacci was a teacher at the Turin Military Academy and had served briefly as an artillery commander with the army during a brief conflict with Austria.11 Rather than calculating the entire flight of a shell, he concentrated on four factors: the range of the trajectory, the time of flight, the maximum height of the shell, and the velocity at impact. Each of these quantities could be used by an artillery officer to plan and direct artillery. The time of flight was used for setting fuses so that the shells would explode over the heads of enemy troops rather than in the ground. The maximum height was used for mortar fire over hills and tall buildings. The terminal velocity gave the amount of energy in a shell and suggested the extent of damage it could produce. Rather than compute these values for all guns and all shells, Siacci chose a strategy that resembled Nevil Maskelyne’s approach to the lunar distance method of navigation. He prepared a set of mathematical tables that described idealized trajectories and then gave rules for transforming values from these tables into the motion of a real shell. The final results were only approximations of the real trajectory, but they were sufficiently accurate for the cannon of the day, including the old smoothbore cannon that had been used in the American Civil War and the new rifled steel barrels that were being produced by Krupp Industries in Prussia.12 Siacci’s ballistics tables were quickly adopted by the armies of the industrialized countries. American officers translated Siacci’s original paper into English less than a year after it appeared in Italian.13
Had the First World War been only a duel of large guns across the fields of Flanders, then Siacci’s method would have sufficed for most of the ballistics calculations. This method had its flaws, but most of them could be fixed with only a little effort. The biggest corrections would have included a refinement of the atmosphere drag function and the introduction of more detail into the mathematical equations. Ballistics officers were now able to measure the velocity of shells with electrical instruments. Their experiments had shown the need to incorporate new factors into the equations, including the density of the atmosphere and the direction of high-level winds. These changes could be handled without a large, permanent computing staff.14 Computers were needed to help with artillery problems that came from the growing use of aircraft. Since Siacci’s tables
presented only a few points of the trajectory, they could not be used for antiaircraft artillery, for bombing, or for aerial combat. Anti-aircraft defense, a problem that became increasingly important during the war, was like duck hunting. The gunnery crew would fire an explosive shell into the air, hoping that it would explode near an oncoming plane. To place that shot close to the plane, the gunners needed to know how fast a shell travels along the upward slope of its trajectory, information that was not easily gleaned from Siacci’s theory.15
The British army first turned to a Cambridge professor, John Littlewood (1885–1977), for help with ballistics analyses. They gave him a lieutenant’s commission and assigned him to work at the Woolwich Arsenal, a military facility located to the east of Greenwich.16 Littlewood was an expert with differential equations and could quickly produce rough approximations of “vertical fire,” the term used by the army for antiaircraft trajectories. When he needed more detailed calculations, he asked for assistance from the faculty and students at Cambridge. Often, he turned to Karl Pearson’s former computer Frances Cave-Browne-Cave at Girton College. Professor Cave-Browne-Cave gladly offered to do calculations for Littlewood, but she had many responsibilities and occasionally needed help herself. “We had to ask my Girton sister to come home before she had finished her work on guns,” reported her sister Beatrice, “so I have been checking some of the most urgent of her work for her.”17
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